how to do this probability cowboy hat problem?

Question: Three tired cowboys entered the saloon and hung their hats on the buffalo horn at the entrance. When the cowboys left in the dead of night, they were unable to distinguish one hat from another, so they took three hats at random. Find the probability that none of them took their own hat.

The answer is 1/3

I got 2/3 as I drew a sample space for all possible combinations of the cowboy and hat so I'm confused

Reply 1

fedora34

You're looking for the scenarios in which none of the cowboys have their own hats.

Let's assign each cowboy a number (1, 2, 3) and each hat a letter (A, B, C). Now let's say that hat A belongs to cowboy 1, hat B to cowboy 2, and hat 3 to cowboy C, so when each cowboy has his own hat it looks like 1A, 2B, 3C.

Now write out each combination

1A, 2B, 3C (they all have their own hat)
1A, 2C, 3B (Cowboy 1 has his own hat)
1B, 2C, 3A (None of them have their own hat)
1B, 2A, 3C (Cowboy 3 has his own hat)
1C, 2A, 3B (None of them have their own hat)
1C, 2B, 3A (Cowboy 2 has his own hat)

So there are six possible combinations and two where none of them have their own hat, so that's 2/6 which is 1/3.